EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 10, No. 2, 2017, 255-271
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global
Nilpotent L-subgroups and the set product of L-subsets
Naseem Ajmal1 , Iffat Jahan2,∗ , Bijan Davvaz3
1
2
3
Department of Mathematics, Zakir Husain College, University of Delhi, Delhi, India
Department of Mathematics, Ramjas College, University of Delhi, Delhi, India
Department of Mathematics, Yazd University, Iran
Abstract. In this paper, we study the set product of a pair of nilpotent normal L-subgroups of
a given L-group. A necessary mechanism is developed in order to establish this result. Moreover,
the tail of L-subgroups is used effectively while developing this mechanism.
2010 Mathematics Subject Classifications: AMS classification codes
Key Words and Phrases: L-algebra; L-subgroup; Generated L-subgroup; Normal L-subgroup;
Nilpotent L-subgroup.
1. Introduction
Rosenfeld [14] applied the notion of fuzzy subsets in algebra and introduced the concept
of fuzzy subgroups in 1971. As a result a new discipline of fuzzy algebraic structures
emerged which contains the extensions of various concepts and notions of classical algebra.
However, the progress of this discipline could not sustain the impact of metatheorem
which was developed by Tom Head [8] during the year 1995. This is due to the fact
that the various notions and concepts formulated in the areas of fuzzy semigroups, fuzzy
groups and fuzzy rings are generically defined and hence the extension of results from
classical algebra to fuzzy algebra became just simple instances of this indigenous result.
Therefore for further growth of the subject, a need was felt to develop a framework for
these investigations which is beyond the purview of the metatheorem. The notion of
lattice valued fuzzy subsets was introduced by Goguen [7] in the year 1967 which was
later applied by Wang Jin Liu [10] to define the notions of lattice valued fuzzy subgroup
of a group and lattice valued ideals of a ring. It is in this framework that the notions
and the concepts extended from classical algebra do not remain projection closed which
is a prerequisite for an application of Tom Head metatheorem. The theory of L-subrings
is sufficiently developed by Mordeson and Malik [13] along with several other researchers
[11, 12]. However, in the area of fuzzy groups such an effort is found lacking. This
∗
Corresponding author.
Email addresses: [email protected] (N. Ajmal), [email protected] (I. Jahan), [email protected]
(B. Davvaz)
http://www.ejpam.com
255
c 2017
EJPAM All rights reserved.
I. Jahan, N. Ajmal, B. Davvaz / Eur. J. Pure Appl. Math, 10 (2) (2017), 255-271
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motivated us to formulate several concepts in the studies of lattice valued fuzzy subgroups
(L-subgroups) which appeared in a series of papers [2, 3, 4, 5, 6]. In all the above mentioned
papers, we not only replaced the evaluation lattice [0,1] by a completely distributive lattice,
we also replaced the parent structure of an ordinary group by an L-group. Therefore
an application of metatheorem in our studies become further remote. Consequently, the
normality of an L-subgroup of an L-group due to Wu has been used in these studies instead
of the concept of normality due to Liu. This allows us to construct the chains of normal
L-subgroups in the same fashion as that of ordinary normal subgroups in classical group
theory. We have introduced and studied the notions of nilpotent L-subgroup, solvable
L-subgroup, normalizer of an L-subgroup and normal closure of an L-group having the
parent structure of an L-group. In the continuation of the development of L-group, the
authors in this paper, after developing a necessary mechanism, prove that the set product
of a pair of nilpotent normal L-subgroups of an L-group is again a nilpotent L-subgroup.
2. Preliminaries
Throughout this paper, the system hL, ≤, ∨, ∧i denotes a completely distributive lattice
where ≤ denotes the partial ordering of L, the join (sup) and meet (inf) of the elements
of L are denoted by ∨ and ∧ respectively. Also, we write 1 and 0 for the maximal and
the minimal elements of L, respectively. Moreover, our work is carried out by using the
definition of L-subset as formulated by Goguen. The definition of a completely distributive
lattice is well known in the literature and can be found in any standard text on the subject.
Let {Ji : i ∈ I} be any family of subsets of a complete
lattice L and F denotes the set
Q
of choice functions for Ji , i.e., functions f : I →
Ji such that f (i) ∈ Ji for each i ∈ I.
i∈I
Then, we say that L is a completely distributive lattice, if
V
W
V W
i∈I f (i) .
f ∈F
i∈I Ji =
The above law is known as the complete distributive law. Moreover, a lattice L is said to
be infinitely meet distributive if for every subset {bβ : β ∈ B} of L, we have
V W
W
V
a {
bβ } =
{a bβ },
β∈B
β∈B
provided L is join complete. The above law is known as the infinitely meet distributive
law. The definition of infinitely join distributive lattice is dual to the above definition i.e.
a lattice L is said to be infinitely join distributive if for every subset {bβ : β ∈ B} of L,
we have
W V
V
a {
bβ } =
{a ∨ bβ },
β∈B
β∈B
provided L is meet complete. The above law is known as the infinitely join distributive
law. Clearly, both these laws follow from the definition of a completely distributive lattice.
Here we also mention that the dual of completely distributive law is valid in a completely
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distributive lattice whereas the infinitely meet and join distributive laws are independent
from each other. Next we recall the following from [1-6, 9, 15]:
An L-subset of X is a function from X into L. The set of L -subsets of X is called the
L-power set of X and is denoted by LX . For µ ∈ LX , the set {µ(x)W: x ∈ X} is called the
image of µ and is denoted by Imµ and the tip of µ is defined as
µ(x). Moreover, the
x∈X
V
tail of µ is defined as
µ(x). We say that an L-subset µ of X is contained in an L-subset
x∈X
η of X if µ(x) ≤ η(x) for x ∈ X and is denoted by µ ⊆ η. For
S a family {µi : i ∈ I} of
L-subsets in X, where I is a nonempty index set, the union
µi and the intersection
i∈I
T
µi of {µi : i ∈ I} are, respectively, defined by:
i∈I
S
µi (x) =
i∈I
W
µ(x) and
i∈I
T
µi (x) =
i∈I
V
µ(x),
i∈I
for each x ∈ X. If µ ∈ LX and a ∈ L, then the notion of level subset µa of µ is defined as:
µa = {x ∈ X : µ(x) ≥ a}.
The set product µ ◦ η of µ, η ∈ LS , where S is a groupoid, is an L-subset of S defined by
W
µ ◦ η(x) =
{µ(y) ∧ η(z)}.
x=yz
Again recall that if x cannot be factored as x = yz in S, then µ◦η(x) being the least upper
bound of the empty set is zero.It can be verified easily that the set product is associative
in LS if S is a semigroup.
Throughout this paper G denotes an ordinary group with the identity element ‘e’, and
I denotes a nonempty indexing set.
Definition 1. Let µ ∈ LG . Then, µ is called an L-subgroup of G if for each x, y ∈ G
(i) µ(xy) ≥ µ(x) ∧ µ(y),
(ii) µ(x−1 ) = µ(x).
The set of L-subgroups of G is denoted by L(G). Clearly, the tip of an L-subgroup is
attained at the identity element e of G.
Definition 2. Let µ ∈ L(G). Then, µ is called a normal L- subgroup of G if, µ(xy) =
µ(yx) for all x, y ∈ G.
It is well known that the intersection of any arbitrary family of L-subgroups of a group
is an L-subgroup of the given group.
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Definition 3. Let µ ∈ LG . Then, the L-subgroup of G generated by µ is defined as the
smallest L-subgroup of G which contains µ. It is denoted by hµi i.e.
hµi = ∩{µi ∈ L(G) : µ ⊆ µi }.
If µ, η ∈ L(G) and η ⊆ µ, then we say that η is an L-subgroup of G. Further, if η is
non-constant and µ 6= η, then η is said to be a proper L-subgroup of µ. Clearly, η is a
proper L-subgroup of µ if and only if η has distinct tip and tail and η 6= µ.
Also, η is said to be a trivial L-subgroup of µ if its chain of level subgroups contains
only e and G. Thus, an L-subgroup may contain several trivial L-subgroups.
Let η be an L-subgroup of µ. Then, we define the following L-subgroup of µ contained
in η, denoted by ηta00 , as follows:
(
a0 , if y = e,
ηta00 (y) =
t0 , if y 6= e,
where a0 = η(e) and t0 = inf η. Here ηta00 , a trivial L-subgroup of µ, is called the trivial
L-subgroup of η.
Henceforth µ denotes an L-subgroup of G and we call the parent L-subgroup simply
an L-group. The set of L-subgroups of µ is denoted by L(µ).
Remark 1. If η ∈ Lµ , then it can be easily verified that hηiµ = hηi, where hηiµ denotes
the L-subgroup of µ generated by η.
We recall the definition of a normal L-subgroup of an L-group.
Definition 4. Let η ∈ L(µ). Then, we say that η is a normal L-subgroup of µ if
η(yxy −1 ) ≥ η(x) ∧ µ(y) for all x, y ∈ G.
The set of normal L-subgroups of µ is denoted by N L(µ).
Proposition 1. Let η ∈ L(µ) and θ ∈ N L(µ). Then,
(i) η ◦ θ ∈ L(µ).
(ii)η ◦ θ ∈ N L(µ) if η ∈ N L(µ).
Proposition 2. Let η, θ ∈ L(µ). Then, η ⊆ η ◦ θ and θ ⊆ η ◦ θ if and only if η(e) = θ(e).
Now, recall the following from [3]:
Definition 5. Let η, θ ∈ Lµ . Then, the commutator of η and θ is an L-subset (η, θ) of G
defined as follows:
(
∨{η(y) ∧ θ(z)}, if x = [y, z] for some y, z ∈ G,
(η, θ)(x) =
inf η ∧ inf θ,
if x 6= [y, z] for any y, z ∈ G.
The commutator L-subgroup of η, θ ∈ Lµ is defined as the L-subgroup of G generated by
(η, θ). It is denoted by [η, θ]. Clearly, inf(η, θ) = inf η ∧ inf θ and [η, θ] ∈ L(µ).
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3. Nilpotent L-subgroup
In [6], Ajmal and Jahan extended the construction of a fuzzy subgroup generated by a
fuzzy subset to L-setting. They proved for an L-subset of a group, the subgroup generated
by its level subset is the level subset of the subgroup generated by that L-subset provided
the given L-subset possesses sup-property. Firstly, we recall from [6] a construction for
generating an L-subgroup by a given L-subset of an L-group.
Theorem 1. Let η ∈ Lµ and a0 = ∨ {η(x)}.Define an L-subset η̂ of G by:
x∈G
η̂(x) = ∨ {a : x ∈ hηa i}.
a≤a0
Then, η̂ ∈ L(µ) and η̂ = hηi.
Proposition 3. Let η, θ ∈ L(µ). Then
[η, θ] (e) = η(e) ∧ θ(e).
Proposition 4. Let η, θ ∈ L(µ) and η ⊆ θ. Then, [η, σ] ⊆ [θ, σ] for each σ ∈ Lµ .
Proposition 5. Let η, θ ∈ N L(µ). Then, [η, θ] ∈ N L(µ).
Next we recall the notion of nilpotent L-subgroup [3]:
Let η ∈ L(µ) and define Z0 (η) = η, Z1 (η) = [Z0 (η), η]. And in general, for each i, we
define Zi (η) = [Zi−1 (η), η].It is easy to verify that Zi (η) ⊆ Zi−1 (η). Moreover, Zi (η) and
η have identical tips and identical tails.
Definition 6. Let η ∈ L(µ) with tip a0 and tail t0 and a0 6= t0 . If the descending central
chain
η = Z0 (η) ⊇ Z1 (η) ⊇ · · · ⊇ Zi (η) ⊇ · · ·
terminates finitely to the trivial L-subgroup ηta00 , then η is known as a nilpotent L-subgroup
of µ. More precisely, η is said to be nilpotent of class c if c is the least non-negative integer
such that Zc (η) = ηta00 . In this case, the series
η = Z0 (η) ⊇ Z1 (η) ⊇ · · · ⊇ Zc (η) = ηta00
is called the descending central series of η. If it is a nilpotent L-subgroup of µ, then we
simply write η is nilpotent.
Proposition 6. Let η ∈ N L(µ). Then, Zi (η) ∈ N L(µ).
Here we give an example of an L-subgroup of an L-group:
Example 1. Let G be the quaternian group Q8 given by :
Q8 = {±1, ±i, ±j, ±k}
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where i2 = j 2 = k 2 = −1, ij = k, jk = i, kj = i . Let C = {1, −1} be the center of G and
H = {±1, ±i}. Let the evaluation lattice L be the chain given by :
L: f ≤a≤b≤d
Define L-subsets µ and η of G as follows:

 d if x ∈ C
b if x ∈ H \ C
µ(x) =

a if x ∈ Q8 \ H.
and

d



b
η(x) =
a



f
if
if
if
if
x=1
x ∈ C \ {1}
x∈H \C
x ∈ Q8 \ H
Since the level subsets of η and µ are subgroups of G, η and µ are L-subgroups of G. As
η ⊆ µ, η is an L-subgroup of µ. Now in view of the fact that every subgroup of a nilpotent
group is nilpotent, it follows that all the level subsets of η are nilpotent subgroups of the
corresponding level subsets of µ. Therefore the converse of Theorem 4.1[3], implies that η
is a nilpotent L-subgroup of µ.
Next we show that just by changing the evaluation lattice L and keeping the parent
group as Q8 , we obtain various types of L-subgroups.
Example 2. Let G = Q8 and the evaluation lattice be given by the diagram :
Consider the parent L-subgroup of G given by:
u if x ∈ C,
µ(x) =
d if x ∈ G \ C.
Now define L-subset η of µ as given below:

u



d
η(x) =
a



b
if
if
if
if
x ∈ C,
x ∈ H1 \ C,
x ∈ H2 \ C,
x ∈ H3 \ C;
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where
C = {±1}, H1 = {±1, ±i}, H2 = {±1, ±j}, H3 = {±1, ±k}.
Since the level subsets of η are normal subgroups of G, η is a normal L-subgroup of G
and hence of µ. Now that η is a nilpotent L-subgroup of µ in view of Definition 3.5.We
demonstrate this as follows:
Note that G0 = {1, −1}. In order to obtain the members of descending central series of η,
we set Z0 (η) = η and consider the commutator (η, η)

 u if x = 1,
d if x ∈ C \ {1},
(η, η) (x) =

f if x ∈ G \ C.
As the level subsets of (η, η) are subgroups of G,
Z1 (η) = [η, η] = (η, η).
Next, we calculate the commutator :
((η, η) , η) (x) =
u if x = 1,
f if x ∈ G \ {1}.
Again by the reasons as given above
Z2 (η) = [[η, η] , η] = ((η, η) , η).
Observe that Z2 (η) is not only an L-subgroup, it is the trivial L-subgroup of η and so the
descending central series terminates at Z2 (η), i. e.
η = Z0 (η) ⊇ Z1 (η) ⊇ Z2 (η) = ηfu .
Consequently η is a nilpotent L−subgroup of µ having nilpotent length 2.
Example 3. Let G = Q8 and the evaluation lattice be given by the diagram :
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Consider the parent L-subgroup of G given by:
u if x ∈ C,
µ(x) =
d if x ∈ G \ C.
Now define L-subsets of µ, η, θ and φ as given below:

d if x ∈ C,



a if x ∈ H1 \ C,
η(x) =
b if x ∈ H2 \ C,



c if x ∈ H3 \ C;
and

d



b
θ(x) =
a



c
if
if
if
if
x ∈ C,
x ∈ H1 \ C,
x ∈ H2 \ C,
x ∈ H3 \ C;

d



a
φ(x) =
c



b
if
if
if
if
x ∈ C,
x ∈ H1 \ C,
x ∈ H2 \ C,
x ∈ H3 \ C;
where
C = {±1}, H1 = {±1, ±i}, H2 = {±1, ±j}, H3 = {±1, ±k}.
Here
ηa = H1 , ηb = H2 and ηc = H3 .
Since the level subsets of η are normal subgroups of G, η is a normal L-subgroup of G.
Similarly, it can be seen that θ and φ are also normal L−subgroups of G and hence of µ.
Now η is a nilpotent L-subgroup of µ, in view of Definition 3.2, can be seen as follows:
Note that G0 = {1, −1}. In order to obtain the members of descending central series of η,
we set Z0 (η) = η and consider the commutator (η, η)
d if x = 1,
(η, η) (x) =
f if x ∈ G \ {1}.
As the level subsets of (η, η) are subgroups of G,
Z1 (η) = [η, η] = (η, η).
Observe that Z1 (η) is not only an L-subgroup, it is the trivial L-subgroup of η and so the
descending central series terminates at Z1 (η), i. e.
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η = Z0 (η) ⊇ Z1 (η) = η0d .
Consequently η is a nilpotent L−subgroup of µ having nilpotent length 1. Now in order to
continue our studies further, we mention that the set product of L-subgroups of η, θ and φ
are given by
η◦θ =θ◦φ=η◦φ=ψ
where ψ is the constant function taking whole of Q8 to d. Obviously η, θ ⊆ ψ ⊆ µ but ψ
being a constant function is not a nilpotent L-subgroup of µ.
Now, we ascertain the tail of the set product of L-subgroups:
Proposition 7. Let η, θ ∈ L(µ). If η(e) = θ(e), then inf η ◦ θ ≥ inf η ∨ inf θ.
The following theorem sufficiently exhibits the application of the notion of infimums:
Theorem 2. Let η, θ ∈ N L(µ) and σ ∈ L(µ). If either η and θ or θ and σ have the same
tails, then
[η ◦ σ, θ] ⊆ [η, θ] ◦ [θ, σ] .
Moreover if η(e) = θ(e), then the equality holds.
Proof. Let x ∈ G. If x is not a commutator and η and θ have the same tails, then
[η, θ] ◦ [θ, σ] (x) ≥ [η, θ] (x) ∧ [θ, σ] (e)
≥ inf η ∧ inf θ ∧ θ(e) ∧ σ(e)
= inf θ ∧ σ(e)
(as inf θ = inf η and inf θ ∧ θ(e) = inf θ)
= inf θ ∧ η(e) ∧ σ(e)
(as inf θ ∧ η(e) = inf η ∧ η(e) = inf η)
= inf θ ∧ η ◦ σ(e)
≥ inf θ ∧ inf η ◦ σ
= (η ◦ σ, θ) (x).
If θ and σ have the same tails, then also
(η ◦ σ, θ) (x) ≤ [η, θ] ◦ [θ, σ] (x).
(1)
Suppose that x is a commutator in G. Now, for any u ∈ G, define the following subsets
of G × G by:
C(u) = {(y, z) ∈ G × G : u = [y, z]} and P (u) = {(y, z) ∈ G × G : u = yz}.
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Now consider
_
(σ ◦ η, θ) (x) =
{σ ◦ η(y) ∧ θ(z)}
(y,z)∈C(x)
_
=
{
_
{σ(u) ∧ η(v)} ∧ θ(z)}
(y,z)∈C(x) (u,v)∈P (y)
_
=
{
_
{σ(u) ∧ η(v) ∧ θ(z)}}
(y,z)∈C(x) (u,v)∈P (y)
_
=
{{σ(u) ∧ θ(z)} ∧ {η(v) ∧ θ(z)}}.
(y,z)∈C(x)
(u,v)∈P (y)
As σ ⊆ µ, we have σ(u) ∧ µ(v) = σ(u). This implies that
_
(σ ◦ η, θ) (x) ≤
{{σ(u) ∧ µ(u) ∧ θ(z)} ∧ {η(v) ∧ θ(z)}}
([v,z]u ,[u,z)∈P (x)
(uv,z)∈C(x)
_
=
{{σ(u) ∧ θ(z)} ∧ {η(v) ∧ θ(z) ∧ µ(u)}}.
([v,z]u ,[u,z])∈P (x)
(uv,z)∈C(x)
Now, we have
[σ, θ] ([u, z]) ≥ σ(u) ∧ θ(z),
and since η, θ ∈ N L(µ), by Proposition 5, [η, θ] ∈ N L(µ). Therefore,
[η, θ] ([v, z]u ) ≥ [η, θ] ([v, z]) ∧ µ(u)
≥ η(v) ∧ θ(z) ∧ µ(u).
Consequently, we have
(σ ◦ η, θ) (x) ≤
_
{[σ, θ] ([u, z]) ∧ [η, θ] ([v, z]u )}
([v,z]u ,[u,z])∈P (x)
(uv,z)∈C(x)
≤ [η, θ] ◦ [σ, θ](x).
(2)
Thus, by (1) and (2)
(σ ◦ η, θ) ≤ [η, θ] ◦ [θ, σ] .
Also, as [η, θ] ∈ N L(µ) and [σ, θ] ∈ L(µ), by Proposition 2.6 [η, θ] ◦ [σ, θ] is an L−subgroup
of µ. Again, by Proposition 2.6, η ◦ σ ∈ L(µ) so that η ◦ σ = σ ◦ η. Hence
[η ◦ σ, θ] = [σ ◦ η, θ] ≤ [η, θ] ◦ [θ, σ] .
Lastly, let η(e) = σ(e). We show that
[η, θ] ◦ [θ, σ] ⊆ [σ ◦ η, θ] .
By Proposition 2, η ⊆ η ◦ θ and σ ⊆ η ◦ θ. By Lemma 5
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265
[η, θ] ⊆ [η ◦ σ, θ] and [σ, θ] ⊆ [η ◦ σ, θ] .
Therefore,
[η, θ] ◦ [θ, σ] ⊆ [η ◦ σ, θ] .
The proof of the following result can be obtained as in classical group theory which
exhibits a routine application of the Principle of Mathematical Induction.
Lemma 1. Let η, η1 , ..., ηn+1 ∈ N L(µ) having identical tails. If ηi = η for k+1 distinct
values of i where 0 ≤ k ≤ n, then [η1 , η2 , ..., ηn+1 ] ⊆ Zk (η).
Next result provides a necessary and sufficient condition for the set product of two
trivial L-subgroups of µ to be a trivial L-subgroup.
Lemma 2. Let η and θ be trivial L-subgroups of µ. Then, the set product η ◦ θ is also a
trivial L-subgroup of µ defined by
(
η(e) ∧ θ(e)
if x = e,
η ◦ θ(x) =
inf η ∨ inf θ if x 6= e,
if and only if inf η ∨ inf θ < η(e) ∧ θ(e).
Proof. Since η and θ are trivial L-subgroups, it follows that
Imη = {inf η, η(e)} and Imθ = {inf θ, θ(e)}.
Thus, if x = e then
η ◦ θ(x) = η(e) ∧ θ(e).
Suppose that x 6= e. If η(e) = a0 ,θ(e) = a∗0 and inf η = t0 , inf θ = t∗0 , then
_
η ◦ θ(x) =
{η(y) ∧ θ(z)}
x=yz
= {η(x) ∧ θ(e)} ∨ {η(e) ∧ θ(x)} ∨ {
∨
b∈G
b6=e,b6=x
η(xb−1 ) ∧ θ(b)}
= {t0 ∧ a∗0 } ∨ {a0 ∧ t∗0 } ∨ {t0 ∧ t∗0 }
= {t0 ∧ a∗0 } ∨ {a0 ∧ t∗0 },
(as a0 ∧ t∗0 ≥ t0 ∧ t∗0 )
= {t0 ∨ {a0 ∧ t∗0 }} ∨ {a∗0 ∨ {a0 ∧ t∗0 }}
= {t0 ∨ {a0 ∧ t∗0 }} ∧ a∗0
(as a∗0 ≥ t∗0 ≥ a0 ∧ t∗0 )
= {a0 ∧ {t0 ∨ t∗0 }} ∧ a∗0
(as L is modular)
= {a0 ∧
a∗0 }
∧ {t0 ∨
t∗0 }.
Thus, if inf η ∨ inf θ < η(e) ∧ θ(e), then η ◦ θ is the trivial L-subgroup given by
(
η(e) ∧ θ(e)
if x = e,
η ◦ θ(x) =
inf η ∨ inf θ if x 6= e.
On the other hand if η ◦ θ is a trivial L-subgroup as given above, then for any x 6= e
I. Jahan, N. Ajmal, B. Davvaz / Eur. J. Pure Appl. Math, 10 (2) (2017), 255-271
266
η ◦ θ(x) = inf η ∨ inf θ = {η(e) ∧ θ(e)} ∧ {inf η ∨ inf θ}.
Thus, inf η ∨ inf θ ≤ η(e) ∧ θ(e). Since η ◦ θ is a trivial L-subgroup
inf η ∨ inf θ 6= η(e) ∧ θ(e).
Therefore, inf η ∨ inf θ < η(e) ∧ θ(e).
Theorem 3. Let η, θ ∈ N L(µ) with common tail t0 such that t0 < η(e) ∧ θ(e) and
inf η ◦ θ = t0 . If η and θ are nilpotent of classes c and d respectively, then η ◦ θ is a
nilpotent L-subgroup of µ of nilpotent class at most c + d.
Proof. Since η, θ ∈ N L(µ), by Proposition 1, we conclude that η ◦ θ ∈ N L(µ). In view
of Proposition 6, Zi (η ◦ θ) ∈ N L(µ). Now, let η and θ be nilpotent of classes c and d
respectively. In order to show that the set product η ◦ θ is nilpotent, we show that the
descending central series of η ◦ θ terminates finitely. Set λ = η ◦ θ so that
inf λ = t0 .
(1)
In view of Lemma 2, the set product of two trivial L-subgroups is a trivial L-subgroup
provided the join of their tails is different from the meet of their tips. Thus, as t0 <
η(e) ∧ θ(e) and by (1), we have
a∗
a ∧a∗0
ηta0o ◦ θt00 = λt0o
,
where ao and a∗0 denote the tips of η and θ respectively. To achieve our aim, we demonstrate
that
a ∧a∗0
Zn (η ◦ θ) = λt0o
,
for some integer n ≥ 0. As η and θ are nilpotent of classes c and d respectively, we get
a∗
Zc (η) = ηta0o and Zd (θ) = θt00 .
(2)
To prove the result, it is sufficient to show that for some positive integer n, Zn (η ◦ θ) is
a∗
contained in the set product of trivial L-subgroups ηta0o and θt00 . Firstly, we claim that for
any positive integer n, Zn (η ◦ θ) is contained in the set product of L-subgroups of the form
[λ1 , λ2 , ..., λn+1 ], where λi = η or θ. As η ◦ θ ∈ N L(µ), in view of (1) and Theorem 2, it
follows that
Z1 (η ◦ θ) = [η ◦ θ, η ◦ θ] ⊆ [η, η] ◦ [η, θ] ◦ [θ, θ].
Suppose that for some positive integer k, Zk (η ◦ θ) is contained in the set product of
L-subgroups of the form [λ1 , λ2 , ..., λk+1 ], where λi = η or θ. Also,
inf Zk (η ◦ θ) = inf η ◦ θ.
I. Jahan, N. Ajmal, B. Davvaz / Eur. J. Pure Appl. Math, 10 (2) (2017), 255-271
267
(3)
Hence in view of (1) and Theorem 2, we have
Zk+1 (η ◦ θ) = [Zk (η ◦ θ), η ◦ θ] ⊆ [Zk (η ◦ θ), η] ◦ [Zk (η ◦ θ), θ].
By the hypothesis Zk (η ◦ θ) is contained in the set product of L-subgroups of the form
[λ1 , λ2 , ..., λk+1 ], where λi = η or θ. Thus, it follows that Zk+1 (η ◦ θ) is contained in the
set product of L-subgroups of the form [λ1 , λ2 , ..., λk+2 ], where λi = η or θ. Thus, by the
Principle of Mathematical Induction our claim is established for every positive integer n.
Now, let n = c + d. Then, in any commutator L-subgroup of the form [λ1 , λ2 , ..., λn+1 ] if
the number of occurrences of η is greater than c, then by Lemma 1 and by (2)
[λ1 , λ2 , ..., λn+1 ] ⊆ Zc (η) = ηta0o .
On the other hand, if the number of occurrences of η is less than or equal to c, then the
number of occurrences of θ is greater than or equal to d + 1. Hence, again by Lemma 1
and by (2)
a∗
[λ1 , λ2 , ..., λn+1 ] ⊆ Zd (θ) = θt0o .
Thus, each L-subgroup of the form [λ1 , λ2 , ..., λn+1 ], where λi = η or θ is contained in
a∗
ηta0o or θt0o . Therefore, Zk (η ◦ θ) is contained in the set product of finitely many trivial
a∗
a∗
a ∧a∗0
L-subgroups ηta0o and θt0o .This product turns out to be ηta0o ◦ θt0o = λt00
hand, Zn (η ◦ θ)(e) = η ◦ θ(e). Also, in view of (1) and (3)
.On the other
inf Zn (η ◦ θ) = t0 .
This implies
a ∧a∗0
λt00
a ∧a∗0
Hence Zn (η ◦ θ) = λt00
a ∧a∗0
⊆ Zn (η ◦ θ) ⊆ λt00
.
.
Below we illustrate the above theorem with the help of an example:
Example 4. Let G = Q8 as in Example 1.Let L be the evaluation lattice given by the
diagram :
I. Jahan, N. Ajmal, B. Davvaz / Eur. J. Pure Appl. Math, 10 (2) (2017), 255-271
268
Let C,H1 , H2 and H3 be the following subgroups of G:
C = {±1}, H1 = {±1, ±i}, H2 = {±1, ±j}, H3 = {±1, ±k}.
Consider the parent L-subgroup of G, defined as follows:
u if x = 1,
µ(x) =
d if x ∈ G \ {1}.
Now define L-subsets η and θ of G, as given below:

 d if x = 1,
a if x ∈ H1 \ {1},
η(x) =

f if x ∈ G \ H1 ;
and

 u if x = 1,
b if x ∈ H2 \ {1},
θ(x) =

f if x ∈ G \ H2 .
Since the level subsets of η are normal subgroups of G, η is a normal L-subgroup of G.
Similarly, it can be seen that θ is also a normal L−subgroup of G and hence of µ. Now η
is a nilpotent L-subgroup of µ , in view of the Definition 3.5 follows as given below:
Note that G0 = C. In order to obtain the members of descending central series, we start
with Z0 (η) = η. Next consider the commutator (η, η):
d if x = 1,
(η, η) (x) =
f if x ∈ G \ {1}.
As the level subsets of (η, η) are subgroups,
Z1 (η) = [η, η] = (η, η).
Note that Z1 (η) is the trivial L-subgroup ηfd of η and so the descending central series
terminates at Z1 (η) i.e.
η = Z0 (η) ⊇ Z1 (η) = ηfd .
Consequently η is a nilpotent L−subgroup of µ having nilpotent length 1. Similarly it can
be shown that θ is also a nilpotent L−subgroup of µ having nilpotent length 1. Next, we
exhibit the set product of L-subgroups η and θ. It can be verified that
η ◦ θ = θ ◦ η = φ,
I. Jahan, N. Ajmal, B. Davvaz / Eur. J. Pure Appl. Math, 10 (2) (2017), 255-271
where φ is the L−subgroup of µ given by

d if



a if
φ(x) =
b if



f if
269
x ∈ C,
x ∈ H1 \ C,
x ∈ H2 \ C,
x ∈ G \ {H1 ∪ H2 }.
Note that here inf η = inf θ = inf η ◦ θ. Hence in view of the above theorem η ◦ θ is a
nilpotent L-subgroup of µ.
The following example exhibits that the condition inf η = inf θ = inf η ◦ θ is only
sufficient:
Example 5. Let G = Q8 as in Example 1.Let L be the evaluation lattice given by the
diagram :
Let C,H1 , H2 and H3 be the following subgroups of G:
C = {±1}, H1 = {±1, ±i}, H2 = {±1, ±j}, H3 = {±1, ±k}.
Consider the parent L-subgroup of G, defined as follows:
u if x = 1,
µ(x) =
d if x ∈ G \ {1}.
Now define L-subsets η, and θ of G, as

d



a
η(x) =
b



f
and

d



a
θ(x) =
b



f
given below:
if
if
if
if
x ∈ C,
x ∈ H1 \ C,
x ∈ H2 \ C,
x ∈ G \ {H1 ∪ H2 };
if
if
if
if
x ∈ C,
x ∈ H1 \ C,
x ∈ H3 \ C,
x ∈ G \ {H1 ∪ H3 }.
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270
Since the level subsets of η are normal subgroups of G, η is a normal L-subgroup of G.
Similarly, it can be seen that θ is also a normal L−subgroup of G and hence of µ. Now
that η is a nilpotent L-subgroup of µ , in view of the Definition 3.5, can be seen as follows:
Note that G0 = C. In order to obtain the members of descending central series, we start
with Z0 (η) = η. Next consider the commutator (η, η):
d if x = 1,
(η, η) (x) =
f if x ∈ G \ {1}.
As the level subsets of (η, η) are subgroups,
Z1 (η) = [η, η] = (η, η).
Note that Z1 (η) is the trivial L-subgroup ηfd of η and so the descending central series
terminates at Z1 (η) i.e.
η = Z0 (η) ⊇ Z1 (η) = ηfd .
Consequently η is a nilpotent L−subgroup of µ having nilpotent length 1. Similarly it can
be shown that θ is also a nilpotent L−subgroup of µ having nilpotent length 1. Next, we
exhibit the set product of L-subgroups η and θ. It can be verified that
η ◦ θ = θ ◦ η = φ,
where φ is the L−subgroup of µ given by
d if x ∈ H1 ,
φ(x) =
b if x ∈ G \ H1 .
Again in view of the converse of Theorem 4.1[3], it follows that φ is a nilpotent L-subgroup
but
inf φ 6= inf η and inf θ.
Acknowledgements
The first author was supported by Emeritus Fellowship of UGC, India during the
course of development of this paper.
References
[1] N. Ajmal and I. Jahan, A study of normal fuzzy subgroups and characteristic fuzzy
subgroups of a fuzzy group, Fuzzy Information and Engineering, 2 : 123-143, 2012.
[2] N. Ajmal and I. Jahan, Normal Closure of an L-subgroup of an L-group, The Journal
of Fuzzy Mathematics, 22 : 115-126, 2014.
REFERENCES
271
[3] N. Ajmal and I. Jahan, Nilpotency and theory of L subgroups of an L-group, Fuzzy
Information and Engineering, 6 : 2014, 1-17.
[4] N. Ajmal and I. Jahan, An L-point characterization of normality and normalizer of
an L-subgroup of an L-group, Fuzzy Information and Engineering, 6 : 147-166, 2014.
[5] N. Ajmal and I. Jahan, Solvable L-subgroup of an L-group, Iranian Journal of Fuzzy
Systems, 12 : 151-161, 2015.
[6] N. Ajmal and I. Jahan, Generated L-subgroup of an L-group, Iranian Journal of
Fuzzy Systems, 12: 129-136, 2015.
[7] J. A. Goguen, L-fuzzy sets, J. Math. Anal. Appl., 18 : 145-174, 1967 .
[8] T. Head, A metatheorem for deriving fuzzy theorems from crisp versions, Fuzzy Sets
and Systems, 73 : 349-358, 1995.
[9] W.J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8 :
133-139, 1982.
[10] W. J. Liu, Operations on fuzzy ideals, Fuzzy Sets and Systems 11 : 31-34, 1983.
[11] D. S. Malik, J. N. Mordeson and P.S. Nair, Fuzzy normal subgroups in fuzzy subgroups, J. Korean Math. Soc., 29 : 1-8, 1992.
[12] L. Martinez, L Fuzzy subgroups of fuzzy groups and fuzzy ideals of fuzzy rings, J.
Fuzzy Math., 3 : 833-849, 1995.
[13] J. N. Mordeson and D. S. Malik, Fuzzy Commutative Algebra, World Scientific,
(1998).
[14] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 : 512-517, 1971.
[15] W. Wu, Normal fuzzy subgroups, Fuzzy Math., 1 : 21-30, 1981.